\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Find the equation of the line perpendicular to \(\displaystyle y = \frac{x}{3} + 3\) that contains \(\displaystyle \left( 8, \ 3\right)\).
The slope of the line is \(\displaystyle m = \frac{1}{3}\) so the perpendicular line has slope \(\displaystyle m=-3\). The equation has the form \(\displaystyle y=-3x+b\). Using the point \(\displaystyle \left( 8, \ 3\right)\) gives the equation \(\displaystyle 3=-3\left(8\right)+b\) Solving for \(\displaystyle b\) gives \(\displaystyle b = 27\). The equation of the perpendicular line is \(\displaystyle y = 27 - 3 x\).
\begin{question}Find the equation of the line perpendicular to $y = \frac{x}{3} + 3$ that contains $\left( 8, \ 3\right)$.
\soln{9cm}{The slope of the line is $m = \frac{1}{3}$ so the perpendicular line has slope $m=-3$. The equation has the form $y=-3x+b$. Using the point $\left( 8, \ 3\right)$ gives the equation $3=-3\left(8\right)+b$ Solving for $b$ gives $b = 27$. The equation of the perpendicular line is $y = 27 - 3 x$. }
\end{question}
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}
\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}
\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}
\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue
\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the equation of the line perpendicular to <img class="equation_image" title=" \displaystyle y = \frac{x}{3} + 3 " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7Bx%7D%7B3%7D%20%2B%203%20" alt="LaTeX: \displaystyle y = \frac{x}{3} + 3 " data-equation-content=" \displaystyle y = \frac{x}{3} + 3 " /> that contains <img class="equation_image" title=" \displaystyle \left( 8, \ 3\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%208%2C%20%5C%20%203%5Cright%29%20" alt="LaTeX: \displaystyle \left( 8, \ 3\right) " data-equation-content=" \displaystyle \left( 8, \ 3\right) " /> . </p> </p><p> <p>The slope of the line is <img class="equation_image" title=" \displaystyle m = \frac{1}{3} " src="/equation_images/%20%5Cdisplaystyle%20m%20%3D%20%5Cfrac%7B1%7D%7B3%7D%20" alt="LaTeX: \displaystyle m = \frac{1}{3} " data-equation-content=" \displaystyle m = \frac{1}{3} " /> so the perpendicular line has slope <img class="equation_image" title=" \displaystyle m=-3 " src="/equation_images/%20%5Cdisplaystyle%20m%3D-3%20" alt="LaTeX: \displaystyle m=-3 " data-equation-content=" \displaystyle m=-3 " /> . The equation has the form <img class="equation_image" title=" \displaystyle y=-3x+b " src="/equation_images/%20%5Cdisplaystyle%20y%3D-3x%2Bb%20" alt="LaTeX: \displaystyle y=-3x+b " data-equation-content=" \displaystyle y=-3x+b " /> . Using the point <img class="equation_image" title=" \displaystyle \left( 8, \ 3\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28%208%2C%20%5C%20%203%5Cright%29%20" alt="LaTeX: \displaystyle \left( 8, \ 3\right) " data-equation-content=" \displaystyle \left( 8, \ 3\right) " /> gives the equation <img class="equation_image" title=" \displaystyle 3=-3\left(8\right)+b " src="/equation_images/%20%5Cdisplaystyle%203%3D-3%5Cleft%288%5Cright%29%2Bb%20" alt="LaTeX: \displaystyle 3=-3\left(8\right)+b " data-equation-content=" \displaystyle 3=-3\left(8\right)+b " /> Solving for <img class="equation_image" title=" \displaystyle b " src="/equation_images/%20%5Cdisplaystyle%20b%20" alt="LaTeX: \displaystyle b " data-equation-content=" \displaystyle b " /> gives <img class="equation_image" title=" \displaystyle b = 27 " src="/equation_images/%20%5Cdisplaystyle%20b%20%3D%2027%20" alt="LaTeX: \displaystyle b = 27 " data-equation-content=" \displaystyle b = 27 " /> . The equation of the perpendicular line is <img class="equation_image" title=" \displaystyle y = 27 - 3 x " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%2027%20-%203%20x%20" alt="LaTeX: \displaystyle y = 27 - 3 x " data-equation-content=" \displaystyle y = 27 - 3 x " /> . </p> </p>